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erdős #852

← #851 · #853 → (packet.json; erdosproblems.com)

Let $d_{n} = p_{n + 1} - p_{n}$ , where $p_{n}$ is the $n$ th prime. Let $h (x)$ be maximal such that for some $n < x$ the numbers $d_{n}, d_{n + 1}, \dots, d_{n + h (x) - 1}$ are all distinct. Estimate $h (x)$ . In particular, is it true that $h (x) > (lo g x)^{c}$ for some constant $c > 0$ , and $h (x) = o (lo g x)?$

Worked, still open.

number theory · open · 0 attempts

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

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unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

* (d_n:=p_{n+1}-p_n) [[nomath]](the $n$th prime gap)[[/nomath]], and * $a(n):=\max{k\ge 1:\ d_n,d_{n+1},\dots,d_{n+k-1}\ \text{are all distinct}}$.

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.

candidate solution ↗

oeis

A001223 — Prime gaps: differences between consecutive primes.1,2,2,4,2,4,2,4,6,2,6,4,2,4,6,6,2,6,4,2,6,4,6,8,4,2,4,2,4,14,4,6,2,10,2,6,6,4,6,6,2,10,2,4,2,12,12,4,2,4,6,2,10,6,6,6,2,A053597 — Let prime(i) = i-th prime (A000040), let d(i) = prime(i+1)-prime(i) (A001223); a(n) = number of distinct numbers among d(n), d(n+1), d(n+2), ... before first duplicate is encountered.2,1,2,2,2,2,3,3,2,3,3,2,3,2,1,2,3,3,3,3,2,3,4,3,2,2,2,3,2,5,4,3,2,3,2,1,2,2,1,3,2,3,2,3,2,1,3,2,3,4,3,3,2,1,1,2,3,5,4,4,A078515 — Numbers n such that A053597(n) sets a new record.1,7,23,30,94,219,279,773,1856,3724,6999,7000,19205,184163,280103,849876,1870722,3570761,4114341,11271072,55282774,682560

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C \geq 0

. Is there an infinite sequence of

n_{i}

i \to \infty lim \frac{p _{n_{i} + 1} - p _{n_{i}}}{lo g n _{i}} = C ?

A001223 #853Let

d_{n} = p_{n + 1} - p_{n}

p_{n}

n

r (x)

be the smallest even integer

t

d_{n} = t

has no solutions for

n \leq x

.Is it true that

r (x) \to \infty

r (x) / lo g x \to \infty

status

open

finding.noted · reviewer:will-blair · 1 day

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